Contents

Idea

A topos may be thought of as a generalized topological space. Accordingly, the notions of

• locally connected space

• locally 2-connected space

• etc. …

• locally contractible space

have analogs for toposes and (∞,1)-toposes

• locally connected topos

• locally n-connected (n,1)-topos

• etc. …

Locally connected topos

Let $E$ be a topos. An object $A\in E$ is called a connected object if ${\mathrm{hom}}_E(A,-)$ preserves finite coproducts. Equivalently, an object $A$ is connected if it is nonempty (noninitial) and cannot be expressed as a coproduct of two nonempty subobjects.

Definition

A Grothendieck topos $E$ is called a locally connected topos is every object $A\in E$ is a coproduct of connected objects $\{A_i{\}}_{i\in I}$, $A={\coprod }_{i\in I}{A}_i$. It follows that the index set $I$ is unique up to isomorphism, and we write

This construction defines a functor ${\Pi }_0:E\to \mathrm{Set}:A\mapsto {\pi }_0(A)$ which is left adjoint to the constant sheaf functor, the left adjoint part of the global section geometric morphism. Thus, for a locally connected topos we have

This left adjoint ${\Pi }_0$ is the lowest degree incarnation of a general construction of homotopy groups in an (∞,1)-topos.

However, this doesn’t mean that essential geometric morphisms are the “relative” analog of locally connected toposes; in general one needs to impose an additional condition, which is automatic in the case of the global sections morphism, to obtain the notion of a locally connected geometric morphism.

Locally connected and connected

A topos $E$ is called a connected topos if the right adjoint $\mathrm{LConst}:\mathrm{Set}\to E$ is a full and faithful functor.

Notice that for a locally connected topos that is also a connected topos the adjunction

exhibits Set as a reflective subcategory of $E$. We may think then of Set as being the localization of $E$ at those morphisms that induce isomorphisms of connected components.

Examples

• For $X$ a topological space, the category of sheaves $\mathrm{Sh}(X):=\mathrm{Sh}(\mathrm{Op}(X))$ is a locally connected topos precisely if $X$ is a locally connected space. The functor ${\Pi }_0$ sends a sheaf $F\in \mathrm{Sh}(X)$ to the set of connected components of the coresponding etale space.

• For $C=$ CartSp the site of Cartesian space with their good open cover coverage, we have that $\mathrm{Sh}(\mathrm{CartSp})$ is locally connected. An arbitrary $X\in \mathrm{Sh}(\mathrm{CartSp})$ is sent to the colimit ${\lim }_{\to }X\in \mathrm{Set}$. If $X$ is a diffeological space or even a smooth manifold, then this is the set of connected components of the underlying topological space.

• Generally, if $C$ is a site such that constant presheaves on $C$ are sheaves, then the left adjoint ${\Pi }_0$ exists and is given by the colimit functor, by the defining property of the colimit as a left Kan extension of presheaves:

  Write $L:\mathrm{PSh}(C)\to \mathrm{Sh}(C)$ for sheafification and assume that constant presheaves are sheaves. Then

  $${\mathrm{Hom}}_{\mathrm{Sh}(C)}(X,L\mathrm{Const}S)\simeq {\mathrm{Hom}}_{\mathrm{PSh}(C)}(X,L\mathrm{Const}S)\simeq {\mathrm{Hom}}_{\mathrm{PSh}(C)}(X,\mathrm{Const}S)\simeq {\mathrm{Hom}}_{\mathrm{Set}}(\underrightarrow{\lim }X,S).$$Hom_{Sh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S) \,.